Research

My research is focused on general relativity in four and higher dimensions and its  most elementary solutions, black holes.  This is of intrinsic interest in mathematical physics, with close connections to Riemannian geometry (e.g. positive mass theorem, Riemannian Penrose inequality,  Einstein metrics on compact manifolds). Further, black holes provide an extreme setting in which quantum effects due to intense gravitational fields become manifest. Much of what is known about quantum gravity has emerged from their study, most strikingly Hawking's formula for the entropy of black holes.

Recent developments in theoretical physics (string theory and the gauge theory/gravity correspondence) attempting to combine quantum mechanics and gravity lead to extra spatial dimensions. A useful first step is to achieve a full understanding of black holes in the associated classical regime (i.e. as geometries satisfying Einstein's equations). The solution to this problem (under certain assumptions, such as analyticity) in four dimensions is a seminal result of mathematical relativity: there is only one kind of equilibrium black hole, which is spherical and uniquely specified by its mass M and angular momentum J. In contrast, the geometry and topology of higher-dimensional black holes is far less constrained and their classification is a formidable task.   Two important recent results in this area are the proof of Galloway and Schoen  that spatial cross-sections of a black hole event horizon must admit a metric with positive curvature and the rigidity theorem proved by Hollands, Ishibashi, and Wald (independently arrived at by Isenberg and Moncrief).

My recent work has concentrated in three main directions. 

Firstly, an important open problem is to achieve a deeper understanding of which horizon topologies actually exist as solutions to Einstein's equations.  In five dimensions, the work of Galloway and Schoen shows the only allowed possibilities are spheres S^3 (and quotients) , rings S^1 X S^2, and connected sums of these.  In recent work with James Lucietti  at the University of Edinburgh, we constructed a smooth asymptotically flat solution describing a black hole with lens-space horizon topology.  This lens space can be through of a Z_2 quotient of the three-sphere.  Along with black rings, these are the only black hole solutions known with non-spherical horizon topology.  We are also investigating the role that non-trivial topology in the region exterior to the event horizon plays in black hole non-uniqueness. 

Secondly, I am interested in geometric inequalities for asymptotically flat Riemannian manifolds admitting isometries which arise naturally in the context of initial data for Einstein's equations.  Such geometries are characterized by certain geometric invariants (e.g. ADM mass M and angular momentum J).  For three-dimensional manifolds, Sergio Dain established the `isoperimetric' result that the mass M is bounded below by |J| with the unique minimizer being the initial data for the extreme (M^2=|J|) Kerr black hole.  I am currently investigating generalisations of this result to higher dimensions.  A first step in this direction is the construction of a mass functional for maximal initial data whose critical points are precisely the geometries arising from stationary, axisymmetric solutions of the vacuum field equations (in collaboration with my former Ph.D student Aghil Alaee who is now a postdoctoral fellow at the University of Alberta). Recently,  in collaboration with  Marcus Khuri of Stony Brook University, we have established a geometric inequality for four-dimensional initial data by exploiting the fact that the mass functional is related to a Dirichlet energy. The critical points are therefore harmonic maps, with the target space being a manifold equipped with a metric of negative curvature.

Thirdly,  a long-term research interest is the classification of near-horizon geometries of extreme black hole. A review article on this area appeared in Living Reviews in Relativity (link to journal page)